Magnetism

37 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

Learning Objectives

  • Describe how magnetic fields exert forces on moving electric charges.
  • Apply the right-hand rule to determine the direction of the magnetic force on a moving charge.
  • Calculate the magnitude of the magnetic force acting on a moving charge.

Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

In the previous chapters, we learned that magnetic fields are produced by permanent magnets and by moving electric charges. But how do magnetic fields interact with other charged particles? The answer is one of the fundamental principles of electromagnetism: magnetic fields exert forces on moving charges.

This idea explains not only why magnets attract or repel one another, but also how electric motors operate, how charged particles are guided inside particle accelerators, and how magnetic fields are used in many medical technologies such as magnetic resonance imaging (MRI). In each of these examples, moving electric charges experience forces that change their motion.

The Magnetic Force

Unlike the electric force, which can act on stationary or moving charges, the magnetic force acts only on charges that are moving. If a charged particle is at rest, a magnetic field exerts no force on it. The faster the particle moves, the larger the magnetic force can become.

The magnitude of the magnetic force acting on a particle with charge q moving at speed v through a magnetic field of strength B is given by

[latex]F=qvB\sin\theta,[/latex]

where θ is the angle between the velocity vector [latex]\mathbf{v}[/latex] and the magnetic field [latex]\mathbf{B}[/latex]. This relationship is commonly called the Lorentz force equation.

This equation reveals several important features of magnetic forces:

  • The force is directly proportional to the magnitude of the charge.
  • The force increases with the particle's speed.
  • Stronger magnetic fields produce larger forces.
  • The force depends on the angle between the particle's motion and the magnetic field.

The angle dependence is especially important. Since the equation contains the factor [latex]\sin\theta[/latex]:

  • If the particle moves parallel or antiparallel to the magnetic field ([latex]\theta=0^\circ[/latex] or [latex]180^\circ[/latex]), then [latex]\sin\theta=0[/latex] and the magnetic force is zero.
  • If the particle moves perpendicular to the magnetic field ([latex]\theta=90^\circ[/latex]), then [latex]\sin\theta=1[/latex] and the magnetic force is at its maximum value:
[latex]F_{\max}=qvB.[/latex]

Unlike electric forces, magnetic forces never act along the direction of motion. Instead, the magnetic force is always perpendicular to both the particle's velocity and the magnetic field. As a result, magnetic fields change the direction of a particle's motion without directly changing its speed.

Because the magnetic force is used to define magnetic field strength, we can rearrange the previous equation to obtain

[latex]B=\frac{F}{qv\sin\theta}.[/latex]

The SI unit of magnetic field strength is the tesla (T), named after the inventor and engineer Nikola Tesla. Using the equation above, we find

[latex]1~\text{T}=\frac{1~\text{N}}{1~\text{C}\cdot\text{m/s}}=\frac{1~\text{N}}{1~\text{A}\cdot\text{m}},[/latex]

since one ampere is equal to one coulomb per second.

Another commonly used unit is the gauss (G), where

[latex]1~\text{G}=10^{-4}\text{ T}.[/latex]

Typical magnetic field strengths vary over many orders of magnitude. The Earth's magnetic field is about [latex]5\times10^{-5}\text{ T}[/latex] (approximately 0.5 G), while strong permanent magnets can produce fields approaching 2 T. Superconducting electromagnets used in research laboratories and MRI scanners routinely generate fields of 3 T or higher, with specialized research systems exceeding 10 T.

The Right-Hand Rule

Knowing the magnitude of the magnetic force is only part of the problem. We also need to determine its direction. Because the force is perpendicular to both the particle's velocity and the magnetic field, we use the right-hand rule.

For a positive charge:

  1. Point your right thumb in the direction of the particle's velocity [latex]\mathbf{v}[/latex].
  2. Point your fingers in the direction of the magnetic field [latex]\mathbf{B}[/latex].
  3. Your palm points in the direction of the magnetic force [latex]\mathbf{F}[/latex].

If the particle carries a negative charge, such as an electron, the magnetic force points in the direction opposite to that given by the right-hand rule.

The right-hand rule for determining the magnetic force on a moving charge. The thumb points in the direction of the particle's velocity, the fingers point in the direction of the magnetic field, and the palm indicates the direction of the magnetic force on a positive charge.
Figure 37.1: The right-hand rule is used to determine the direction of the magnetic force on a positive moving charge. The force is always perpendicular to both the velocity and the magnetic field. For a negative charge, the force points in the opposite direction.

Making Connections: Moving Charges Create and Respond to Magnetic Fields

A stationary electric charge produces an electric field but experiences no magnetic force. Once the charge begins moving, however, it both generates a magnetic field and can experience magnetic forces produced by other magnetic fields. This close relationship between electricity and magnetism is one of the central ideas of electromagnetism and ultimately led to Maxwell's unified theory of electromagnetic fields.

Example 37.1: Calculating Magnetic Force on a Charged Object in Earth's Magnetic Field

Although Earth's magnetic field is relatively weak, it still exerts forces on moving electric charges. In everyday life these forces are far too small to notice on ordinary objects, but they become important for electrons, ions, and other microscopic charged particles.

Suppose a glass rod is rubbed with silk, giving it a positive charge of 20.0 nC. The rod is then thrown horizontally toward the west with a speed of 10.0 m/s. At that location, Earth's magnetic field has a magnitude of [latex]5.0\times10^{-5}\text{ T}[/latex] and points due north, parallel to the ground. Calculate the magnitude of the magnetic force acting on the rod. Determine the direction of the force using the right-hand rule.

A positively charged object moves due west while Earth's magnetic field points due north. The magnetic force acts downward according to the right-hand rule. The accompanying illustration shows the right hand with the thumb pointing west, the fingers pointing north, and the palm indicating the downward direction of the magnetic force.
Figure 37.2: A positively charged object moving due west in Earth's magnetic field experiences a magnetic force directed downward. A negatively charged object moving in the same direction would experience an upward force.

Strategy

The magnitude of the magnetic force is given by

[latex]F=qvB\sin\theta.[/latex]

The velocity points west while the magnetic field points north, so the angle between them is [latex]90^\circ[/latex]. Therefore, [latex]\sin\theta=1[/latex], and the force has its maximum possible value.

Solution

Substitute the known values into the magnetic force equation:

[latex]\begin{aligned} F &= qvB\sin\theta\\ &=(20.0\times10^{-9}\,\text{C})(10.0\,\text{m/s})(5.0\times10^{-5}\,\text{T})(1)\\ &=1.0\times10^{-11}\,\text{N}. \end{aligned}[/latex]

To determine the direction, apply the right-hand rule:

  • Point your thumb toward the west (the direction of the velocity).
  • Point your fingers toward the north (the direction of the magnetic field).
  • Your palm points downward, indicating the direction of the magnetic force on a positive charge.

Therefore, the magnetic force has a magnitude of

[latex]F=1.0\times10^{-11}\,\text{N},[/latex]

and it is directed vertically downward.

Discussion

A force of [latex]1.0\times10^{-11}\text{ N}[/latex] is extraordinarily small and has no noticeable effect on a macroscopic object such as a charged glass rod. This agrees with everyday experience—we do not observe Earth's magnetic field significantly altering the motion of objects around us.

The situation is very different for microscopic charged particles such as electrons and ions. Because they have extremely small masses, even modest magnetic forces can dramatically change their trajectories. This principle is fundamental to technologies such as particle accelerators, mass spectrometers, electron microscopes, and many medical imaging techniques.

Section Summary

  • A magnetic field exerts a force only on moving electric charges. The magnitude of the magnetic force is given by
    [latex]F=qvB\sin\theta,[/latex]

    where [latex]\theta[/latex] is the angle between the particle's velocity [latex]\mathbf{v}[/latex] and the magnetic field [latex]\mathbf{B}[/latex].

  • The SI unit of magnetic field strength is the tesla (T), defined as
    [latex]1~\text{T}=\frac{1~\text{N}}{1~\text{C}\cdot\text{m/s}}=\frac{1~\text{N}}{1~\text{A}\cdot\text{m}}.[/latex]
  • The direction of the magnetic force on a positive charge is determined using the right-hand rule: point your thumb in the direction of the particle's velocity, your fingers in the direction of the magnetic field, and your palm indicates the direction of the magnetic force. For a negative charge, the force is in the opposite direction.
  • The magnetic force is always perpendicular to both the particle's velocity and the magnetic field. Consequently, magnetic fields change the direction of a charged particle's motion without directly changing its speed.
  • The magnetic force is zero when a charged particle moves parallel or antiparallel to the magnetic field and is greatest when the motion is perpendicular to the field.

Conceptual Questions

6. If a charged particle moves in a straight line through some region of space, can you conclude that the magnetic field in that region must be zero? Explain your reasoning.

Problems & Exercises

  1. Determine the direction of the magnetic force acting on a positive charge moving in each of the six situations shown in Figure 37.3.
    Six diagrams showing different combinations of magnetic field directions and charge velocities for determining the direction of the magnetic force using the right-hand rule.
    Figure 37.3.
  2. Repeat Problem 1 for a negative charge.
  3. Determine the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in Figure 37.4, assuming the velocity is perpendicular to the magnetic field.
    Three diagrams showing magnetic force and magnetic field directions. Students determine the velocity direction of a negative charge using the right-hand rule.
    Figure 37.4.
  4. Repeat Problem 3 for a positive charge.
  5. Determine the direction of the magnetic field that produces the magnetic force on a positive charge in each of the three situations shown in Figure 37.5, assuming the magnetic field is perpendicular to the velocity.
    Three diagrams showing the velocity and magnetic force acting on a positive charge. Students determine the magnetic field direction using the right-hand rule.
    Figure 37.5.
  6. Repeat Problem 5 for a negative charge.
  7. An aluminum rod carries a net charge of [latex]0.100~\mu\text{C}[/latex] and is moved at a speed of [latex]5.00~\text{m/s}[/latex] between the poles of a [latex]1.50~\text{T}[/latex] permanent magnet. Assuming the force is maximum, calculate the magnitude of the magnetic force acting on the rod. In what direction does the force act?
  8. Aircraft can accumulate small amounts of static electric charge during flight.
    1. Suppose a supersonic jet carries a charge of [latex]0.500~\mu\text{C}[/latex] and flies due west at a speed of [latex]660~\text{m/s}[/latex] over Earth's south magnetic pole, where the magnetic field has a magnitude of [latex]8.00\times10^{-5}~\text{T}[/latex] and points vertically upward. Determine the magnitude and direction of the magnetic force on the aircraft.
    2. Based on your result, discuss whether this magnetic force is significant for the motion of the aircraft.
  9. A cosmic-ray proton moving toward Earth at [latex]5.00\times10^{7}~\text{m/s}[/latex] experiences a magnetic force of [latex]1.70\times10^{-16}~\text{N}[/latex].
    1. If the angle between the proton's velocity and the magnetic field is [latex]45^\circ[/latex], determine the magnetic field strength.
    2. Is your answer consistent with the known strength of Earth's magnetic field near its surface? Explain.
  10. An electron moving at [latex]4.00\times10^{3}~\text{m/s}[/latex] in a magnetic field of magnitude [latex]1.25~\text{T}[/latex] experiences a magnetic force of [latex]1.40\times10^{-16}~\text{N}[/latex]. Determine the angle between the electron's velocity and the magnetic field. There are two possible answers.
  11. A physicist performing a sensitive experiment wants to ensure that the magnetic force on a moving charged object remains below [latex]1.00\times10^{-12}~\text{N}[/latex].
    1. Assuming the object moves at no more than [latex]30.0~\text{m/s}[/latex] in Earth's magnetic field, determine the greatest allowable charge.
    2. Compare your answer with the magnitude of typical static electric charges and discuss whether maintaining such a small charge would be practical.

Glossary

right-hand rule
A rule used to determine the direction of the magnetic force on a positive moving charge: point your right thumb in the direction of the particle's velocity and your fingers in the direction of the magnetic field. Your palm points in the direction of the magnetic force. For a negative charge, the force is in the opposite direction.
Lorentz force
The magnetic force exerted on a charged particle moving through a magnetic field.
tesla (T)
The SI unit of magnetic field strength, defined as [latex]1~\text{T}=\dfrac{1~\text{N}}{1~\text{A}\cdot\text{m}}.[/latex]
magnetic force
The force exerted on a moving electric charge by a magnetic field. Its magnitude depends on the charge, its speed, the magnetic field strength, and the angle between the velocity and the magnetic field.
gauss (G)
A unit of magnetic field strength commonly used in some scientific and engineering applications, where [latex]1~\text{G}=10^{-4}~\text{T}.[/latex]
definition

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Introductory Physics for the Health and Life Sciences II Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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